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```1917]
ON  PERIODIC  IRROTATIONAL WAVES
479
permanent waves of moderate height do exist. If this is so, .and if Stokes' series are convergent in the mathematical sense for such heights, it appears very unlikely that the case will be altered until the wave attains the greatest admissible elevation, when, as Stokes showed, the crest comes to an edge at an angle of 120°.
It may be remarked that most of the authorities mentioned above express belief in the existence of permanent waves, even though the water be not deep, provided of course that the bottom be flat. A further question may be raised as to whether it is necessary that gravity be constant at different levels. In the paper first cited I showed that, under a gravity inversely as the cube of the distance from the bottom, very long waves are permanent. It may be that under a wide range of laws of gravity permanent waves exist.
Following the method of my paper of 1911, we suppose for brevity that the wave-length is 2-Tr, the velocity of propagation unity*, and we take as the expression for the stream-function of the waves, reduced to rest,
^ = y — ae~v cos x — /3e~22/ cos 2# — <ye~31J cos 3#
— Se~4y cos 4# — ee~5i/ cos 5w,    ...... (1 )
in which x is measured horizontally and y vertically downwards. This expression evidently satisfies the differential equation to which -v/r is subject, whatever may be the values of the constants a, /3, &c. And, much as before, we shall find that 'the surface condition can be satisfied to the order of a7 inclusive ; /3, 7, 8, e being respectively of orders a4, a5, a", a7.
We suppose that the free surface is the stream-line ^ = 0, and the constancy of pressure there imposed requires the constancy of Uz — 2gy, where U, representing the resultant velocity, is equal to ^/{(d^/dan)2' + (cfyr/<%)2}> and g is the constant acceleration of gravity now to be determined. Thus when ^ = 0,
U2 - 2gy = 1 + 2 (1 - g) y + cPe'*" + 2/3e~^ cos 2a?
4- 4ye~32/ cos 2>x + QSe~*v cos 4» + 8ee~&v cos 5sc
+ 4<afte~3v cos x + 6aye~*y cos Zx + 8a&e~5v cos Bay   ......... (2)
correct to a7 inclusive. On the right of (2) we have to expand the exponentials and substitute for the various powers of y expressions in terms of x. »
It may be well to reproduce the process as formerly given, omitting S and e, and carrying (2) only to the order a". We have from (1) as successive approximations to y. —
y = ae~y cos X — CL cos'x; ........................... (8)
* The extension to arbitrary wave-lengths and velocities may be effected at any time by attention to dimensions.resistance, the dissipation function takes the place of the energy function. If an electromotive force act on any branch of a network of conductors, it gt-nerates less current, and accordingly does less work, when an interruption occurs, as by breaking a contact in any part of the system.f gelatine, scarcely thicker than thick paper, should be able to tear out fragments of-solid glass, but there is no doubt of the fact.
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